[Paper Review] Spontaneous Symmetry Breaking of Lorentz and (Galilei) Boosts in (Relativistic) Many-Body Systems

Manfred Requardt|arXiv (Cornell University)|May 20, 2008

Quantum Mechanics and Applications25 references6 citations

TL;DR

This paper demonstrates that spontaneous symmetry breaking (SSB) of Lorentz and Galilei boosts occurs in relativistic and non-relativistic many-body systems due to non-vanishing particle or energy density, leading to gapless Goldstone excitations of phonon-like character. It shows that these modes emerge via a redefined Hamiltonian in the thermodynamic limit, with energy-momentum spectra covering the full positive half-space, confirming a generalized Goldstone phenomenon despite the absence of time-translation invariance for boost generators.

ABSTRACT

We extend a result by Ojima on spontaneous symmetry breaking of Lorentz boosts in thermal (KMS) states and show that it is in fact a special case in a more general class of examples of spontaneous symmetry breaking of Lorentz symmetry in relativistic many-body systems. Furthermore we analyse the nature of the corresponding Goldstone phenomenon and the type of Goldstone excitations (provided they have particle character).

Motivation & Objective

• To generalize Ojima's result on SSB of Lorentz boosts in KMS states to a broader class of relativistic and non-relativistic many-body systems.
• To analyze the nature of Goldstone excitations arising from boost SSB, particularly their particle-like character and gapless dispersion.
• To clarify the conditions under which a Goldstone theorem holds despite the non-commutativity of boost generators with time evolution.
• To distinguish collective excitations (including Goldstone modes) from quasi-particles and vacuum particles in finite-volume and thermodynamic limits.

Proposed method

• Introduces a spectral support analysis of operators and states to handle non-translation-covariant currents, particularly those associated with Lorentz and Galilei boosts.
• Defines a renormalized Hamiltonian $ K = H - n ho_0 $, where $ \rho_0 $ is the energy or particle density, to isolate the physical excitation spectrum.
• Uses Fock space formalism in finite volume with $ n $-particle states to study the action of boost-related creation/annihilation operators $ b^\dagger(\mathbf{k}) $, $ b(\mathbf{k}) $.
• Analyzes the energy spectrum of $ N $-particle excitations in the thermodynamic limit, showing that low-momentum modes can yield arbitrarily small energy for fixed total momentum.
• Applies a scaling argument with $ \varepsilon_i \sim \varepsilon/4 $, $ N \sim 2k/\varepsilon $, to demonstrate that $ E_N \to 0 $ as $ N \to \infty $, even with fixed total momentum $ \mathbf{k} $.
• Treats zero-momentum modes as c-numbers in the thermodynamic limit, with $ a(0)/n^{1/2} \to e^{i\alpha} \delta(\mathbf{k}) $, to handle condensate effects and gauge symmetry breaking.

Experimental results

Research questions

• RQ1 Under what conditions does spontaneous symmetry breaking of Lorentz boosts occur in many-body systems, and how does it generalize beyond thermal KMS states?
• RQ2 How do Goldstone excitations manifest in systems where the boost generator does not commute with the Hamiltonian, challenging standard Goldstone theorems?
• RQ3 What is the nature of the energy-momentum spectrum of the resulting excitations, and does it cover the full positive half-space in the thermodynamic limit?
• RQ4 How do collective excitations, particularly phonon-like modes, emerge from the interplay between relativistic dynamics and broken boost symmetry?
• RQ5 What is the role of the renormalized Hamiltonian $ K = H - \mu \hat{N} $ in defining the physical spectrum and enabling the Goldstone mode analysis?

Key findings

• Spontaneous symmetry breaking of Lorentz and Galilei boosts arises due to non-vanishing particle or energy density, generalizing Ojima’s KMS-state result to broader many-body systems.
• Goldstone excitations are gapless and of phonon-type, with dispersion $ \omega_k - \omega_0 \approx \frac{1}{2m} k^2 $ for small $ k $, indicating a soft mode spectrum.
• In the thermodynamic limit, the energy of $ N $-particle excitations with fixed total momentum $ \mathbf{k} $ can be made arbitrarily small, satisfying $ E_N \to 0 $ as $ N \to \infty $, confirming the existence of gapless modes.
• The redefined Hamiltonian $ K = H - \mu \hat{N} $, with $ \mu = \omega_0 $, isolates the physical spectrum and allows the identification of gapless modes via $ K b^\dagger(\mathbf{k}) |n\rangle = (\omega_k - \omega_0) b^\dagger(\mathbf{k}) |n\rangle $.
• Zero-momentum modes behave as c-numbers in the thermodynamic limit, with $ a(0)/n^{1/2} \to e^{i\alpha} \delta(\mathbf{k}) $, reflecting condensate formation and gauge symmetry breaking.
• The full positive energy-momentum spectrum is covered in the thermodynamic limit, indicating that the system supports gapless excitations for all momenta, consistent with a generalized Goldstone phenomenon.